Unary coding

Unary coding,^{[nb 1]} or the unary numeral system and also sometimes called thermometer code, is an entropy encoding that represents a natural number, n, with a code of length n + 1 ( or n ), usually n ones followed by a zero (if natural number is understood as non-negative integer) or with n − 1 ones followed by a zero (if natural number is understood as strictly positive integer). For example 5 is represented as 111110 or 11110. Some representations use n or n − 1 zeros followed by a one. The ones and zeros are interchangeable without loss of generality. Unary coding is both a prefix-free code and a self-synchronizing code.

Unary coding is an optimally efficient encoding for the following discrete probability distribution

${\displaystyle \operatorname {P} (n)=2^{-n}\,}$

for ${\displaystyle n=1,2,3,...}$.

In symbol-by-symbol coding, it is optimal for any geometric distribution

${\displaystyle \operatorname {P} (n)=(k-1)k^{-n}\,}$

for which k ≥ φ = 1.61803398879..., the golden ratio, or, more generally, for any discrete distribution for which

${\displaystyle \operatorname {P} (n)\geq \operatorname {P} (n+1)+\operatorname {P} (n+2)\,}$

for ${\displaystyle n=1,2,3,...}$. Although it is the optimal symbol-by-symbol coding for such probability distributions, Golomb coding achieves better compression capability for the geometric distribution because it does not consider input symbols independently, but rather implicitly groups the inputs. For the same reason, arithmetic encoding performs better for general probability distributions, as in the last case above.
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